Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 915739, 13 pages doi:10.1155/2010/915739 Research Article Existence of Solutions and Nonnegative Solutions for Prescribed Variable Exponent Mean Curvature Impulsive System Initialized Boundary Value Problems Guizhen Zhi,1 Liang Zhao,2 Guangxia Chen,3 Xiaopin Liu,1 and Qihu Zhang1, 4 1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China 2 Academic Administration, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China 3 School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, Henan 454003, China 4 School of Mathematics and Statistics, Huazhong Normal University, Wuhan, Hubei 430079, China Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn Received 10 May 2009; Accepted 23 January 2010 Academic Editor: Ondřej Dosly Copyright q 2010 Guizhen Zhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the existence of solutions and nonnegative solutions for prescribed variable exponent mean curvature impulsive system initialized boundary value problems. The proof of our main result is based upon Leray-Schauder’s degree. The sufficient conditions for the existence of solutions and nonnegative solutions have been given. 1. Introduction The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. On the Laplacian impulsive differential equations boundary value problems, there are many results see 1–5. Because of the nonlinearity of p-Laplacian, the results about p-Laplacian impulsive differential equations boundary value problems are rare see 6. In 7, 8, the authors discussed the existence of solutions of pr-Laplacian system impulsive boundary value problems. Recently, the existence and asymptotic behavior of solutions of curvature equations have been studied extensively see 9–15. In 16, the authors generalized the usual mean curvature systems to variable exponent mean curvature systems. In this paper, we consider the existence of 2 Journal of Inequalities and Applications solutions and nonnegative solutions for the prescribed variable exponent mean curvature system − ϕ r, u f r, u, u 0, 1.1 r ∈ 0, T , r / ri , where u : 0, T → RN , with the following impulsive initialized boundary value conditions lim ur − lim− ur Ai r → ri r → ri lim ur, lim− u r , r → ri− lim ϕ r, u r − lim− ϕ r, u r Bi r → ri r → ri 1.2 r → ri i 1, . . . , k, lim ur, lim− u r , r → ri− r → ri i 1, . . . , k, u 0 u0 0, 1.3 1.4 where |x|pr−1 x 1/qr , 1 |x|qrpr ϕr, x ∀r ∈ 0, T , x ∈ RN , 1.5 p, q ∈ C0, T , R are absolutely continuous, where p, q satisfy pr ≥ 1, qr ≥ 1, −ϕr, u is called the variable exponent mean curvature operator, 0 < r1 < r2 < · · · < rk < T and Ai , Bi ∈ CRN × RN , RN . For any v ∈ RN , vj will denote the jth component of v; the inner product in RN will be denoted by ·, ·; | · | will denote the absolute value and the Euclidean norm on RN . Denote that J 0, T , J 0, T \ {r0 , r1 , . . . , rk1 }, and J0 r0 , r1 , Ji ri , ri1 , i 1, . . . , k, where r0 0, rk1 T . Denote that Jio is the interior of Ji , i 0, 1, . . . , k. Let ⎧ ⎫ x ∈ C Ji , RN , i 0, 1, . . . , k, ⎬ ⎨ PC J, RN x : J −→ RN , ⎩ lim xr exists for i 1, . . . , k⎭ r → ri ⎫ x ∈ C Jio , RN , ⎬ . PC1 J, RN x ∈ PC J, RN x r and lim− x r exist for i 0, 1, . . . , k⎭ ⎩ rlim → r r →r ⎧ ⎨ i 1.6 i1 For any ur u1 r, . . . , uN r ∈ PCJ, RN , denote that |ui |0 sup{|ui r| | r ∈ J }. i 2 1/2 , and PC1 J, RN Obviously, PCJ, RN is a Banach space with the norm u0 N i1 |u |0 is a Banach space with the norm u1 u0 u 0 . In the following, PCJ, RN and PC1 J, RN will be simply denoted by PC and PC1 , respectively. Denote that L1 L1 J, RN , T i 2 1/2 and the norm in L1 is uL1 N . i1 0 |u r|dr The study of differential equations and variational problems with variable exponent conditions is a new and interesting topic. For the applied background on this kind of problems we refer to 17–19. Many results have been obtained on this kind of problems, Journal of Inequalities and Applications 3 for example, 20–35. If pr ≡ p a constant and qr ≡ q a constant, then 1.1 is the wellknown mean curvature system. Since problems with variable exponent growth conditions are more complex than those with constant exponent growth conditions, many methods and results for the latter are invalid for the former; for example, if Ω ⊂ Rn is a bounded domain, the Rayleigh quotient λpx inf 1,px u∈W0 Ω\{0} 1/px |∇u|px dx 1/px |u|px dx Ω Ω 1.7 is zero in general, and only under some special conditions λpx > 0 see 25, but the fact that λp > 0 is very important in the study of p-Laplacian problems. In this paper, we investigate the existence of solutions for the prescribed variable exponent mean curvature impulsive differential system initialized boundary value problems; the proof of our main result is based upon Leray-Schauder’s degree. This paper was motivated by 6, 13, 36. Let N ≥ 1, then the function f : J × RN × RN → RN is assumed to be Caratheodory; by this we mean that i for almost every t ∈ J the function ft, ·, · is continuous, ii for each x, y ∈ RN × RN the function f·, x, y is measurable on J, iii for each R > 0 there is a βR ∈ L1 J, R such that, for almost every t ∈ J and every x, y ∈ RN × RN with |x| ≤ R, |y| ≤ R, one has f t, x, y ≤ βR t. 1.8 We say a function u : J → RN is a solution of 1.1 if u ∈ PC1 with ϕr, u absolutely continuous on Jio , i 0, 1, . . . , k, which satisfies 1.1 a.e. on J. This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions for system 1.1–1.4. 2. Preliminary In this section, we will do some preparation. Lemma 2.1 see 16. ϕ is a continuous function and satisfies the following. i For any r ∈ J, ϕr, · is strictly monotone, that is, ϕr, x1 − ϕr, x2 , x1 − x2 > 0, for any x1 , x2 ∈ RN , x1 / x2 . 2.1 ii For any fixed r ∈ J, ϕr, · is a homeomorphism from RN to E x ∈ RN | |x| < 1 . 2.2 4 Journal of Inequalities and Applications For any r ∈ J, denote by ϕ−1 r, · the inverse operator of ϕr, ·, then −1/prqr ϕ−1 r, x 1 − |x|qr |x|1/pr−1 x, for x ∈ E \ {0}, ϕ−1 r, 0 0. 2.3 Let one now consider the following simple problem: ϕ r, u r hr, 2.4 r ∈ 0, T , r / ri , with the following impulsive boundary value conditions: lim ur − lim− uri ai , r → ri i 1, . . . , k, r → ri lim ϕ r, u r − lim− ϕ r, u r bi , r → ri r → ri i 1, . . . , k, 2.5 u 0 u0 0, where ai , bi ∈ RN , ki1 |bi | < 1; h ∈ L1 . Obviously, u 0 0 implies that ϕ0, u 0 0. If u is a solution of 2.4 with 2.5, by integrating 2.4 from 0 to r, then one finds that bi ϕ r, u r r ∀r ∈ J . htdt, 2.6 0 ri <r Define a a1 , . . . , ak ∈ RkN , b b1 , . . . , bk ∈ RkN , and operator F : L1 → PC as Fhr r htdt, 2.7 ∀h ∈ L1 . 0 From the definition of ϕ, one can see that sup bi Fhr < 1. r∈J r <r 2.8 i Denote that h ∈ L sup bi Fhr < 1 . r∈J r <r Db 1 2.9 i By 2.6, one has ur ri <r −1 ai F ϕ r, ri <r bi Fh r, ∀r ∈ J. 2.10 Journal of Inequalities and Applications 5 Denote that W R2kN × L1 with the norm wW k k |ai | i1 |bi | hL1 , ∀w a, b, h ∈ W, 2.11 i1 then W is a Banach space. Let one define the nonlinear operator Kb : Db → PC1 as −1 Kb hr F ϕ r, bi Fh r, ∀r ∈ J. 2.12 ri <r Lemma 2.2. The operator Kb is continuous and sends closed equiintegrable subsets of Db into relatively compact sets in PC1 . Proof. It is easy to check that Kb h· ∈ PC1 , for all h ∈ Db . Since t, Kb h t ϕ bi Fh , −1 ∀t ∈ J, 2.13 ri <t it is easy to check that Kb is a continuous operator from Db to PC1 . Let now U be a closed equiintegrable set in Db , then there exists η ∈ L1 , such that, for any u ∈ U, |ut| ≤ ηt 2.14 a.e. on J. We want to show that Kb U ⊂ PC1 is a compact set. Let {un } is a sequence in Kb U, then there exists a sequence {hn } ∈ U such that un Kb hn . For any t1 , t2 ∈ J, we have that t2 ηtdt. |Fhn t1 − Fhn t2 | ≤ t1 2.15 Hence the sequence {Fhn } is uniformly bounded and equicontinuous. By AscoliArzela theorem, there exists a subsequence of {Fhn } which we rename the same that is convergent in PC. Then the sequence bi Fhn ϕ t, Kb hn t 2.16 ri <t is convergent according to the norm in PC. Since −1 Kb hn t F ϕ t, bi Fhn ri <t t, ∀t ∈ J, 2.17 6 Journal of Inequalities and Applications according to the continuity of ϕ−1 , we can see that Kb hn is convergent in PC. Thus we conclude that {un } is convergent in PC1 . This completes the proof. We denote that Nf u : PC1 → L1 is the Nemytski operator associated to f defined by Nf ur f r, ur, u r , 2.18 a.e. on J. Define K : PC1 → PC1 as −1 Kur F ϕ r, Bi F Nf u r, ∀r ∈ J, 2.19 ri <r where Bi Bi limr → ri− ur, limr → ri− u r. Lemma 2.3. u is a solution of 1.1–1.4 if and only if u is a solution of the following abstract equation: u Ai Ku, 2.20 ri <r where Ai Ai limr → ri− ur, limr → ri− u r, Bi Bi limr → ri− ur, and limr → ri− u r. Proof. i If u is a solution of 1.1–1.4, since u 0 0 implies that ϕ0, u 0 0, by integrating 1.1 from 0 to r, then we find that Bi ϕ r, u r r f t, u, u dt, ∀r ∈ J . 0 ri <r 2.21 Thus u Ai Ku. 2.22 ri <r Hence u is a solution of 2.20. ii If u is a solution of 2.20, then it is easy to see that 1.2 is satisfied. Let r 0, then we have u0 0. 2.23 From 2.20 we also have Bi F Nf u r, ϕ r, u r ∈ 0, T , r / ri , ri <r ϕr, u f r, u, u , r ∈ 0, T , r / ri . 2.24 2.25 Journal of Inequalities and Applications 7 From 2.24, we can see that 1.3 is satisfied. Let r 0, from 2.24, then we have ϕ 0, u 0 0. 2.26 Since ϕr, x |x|pr−1 x/1 |x|qrpr 1/qr 0, we must have x 0; thus, u 0 0. 2.27 Hence u is a solution of 1.1–1.4. This completes the proof. 3. Main Results and Proofs In this section, we will apply Leray-Schauder’s degree to deal with the existence of solutions for 1.1–1.4. We assume that H0 ki1 |Bi u, v| < 1/2, for all u, v ∈ RN × RN . Theorem 3.1. If (H0 ) is satisfied, then 0 ∈ Ω is an open bounded set in PC1 such that the following conditions hold. 10 For any u ∈ Ω, the mapping r → fr, u, u belongs to {v ∈ L1 | vL1 < 1/3}. 20 For each λ ∈ 0, 1, the problem λf r, u, u , ϕ r, u lim ur − lim− ur λAi r → ri r → ri lim ur, lim− u r , r → ri− lim ϕ r, u r − lim− ϕ r, u r λBi r → ri r ∈ 0, T , r / ri , r → ri 3.1 r → ri i 1, . . . , k, lim ur, lim− u r , r → ri− r → ri i 1, . . . , k, u 0 u0 0 has no solution on ∂Ω. Then 1.1–1.4 has at least one solution on Ω. Proof. Denote that Ai Ai lim ur, lim− u r , r → ri− r → ri Bi Bi Denote that Ψu, λ : λ ri <r Ai Kλ u. 3.2 ∀r ∈ J. 3.3 lim ur, lim− u r . r → ri− r → ri For any λ ∈ 0, 1, define Kλ : PC1 → PC1 as −1 λBi F λNf u Kλ ur F ϕ r, r, ri <r 8 Journal of Inequalities and Applications We know that 1.1–1.4 has the same solutions of u Ψu, 1 Ai K1 u. ri <r 3.4 Since f is Caratheodory, it is easy to see that Nf · is continuous and sends bounded sets into equiintegrable sets. According to Lemma 2.2, we can conclude that Ψ is compact continuous on Ω for any λ ∈ 0, 1. We assume that for λ 1 3.4 does not have a solution on ∂Ω; otherwise, we complete the proof. Now from hypothesis 20 , it follows that 3.4 has no solution for u,λ ∈ ∂Ω × 0, 1. When λ 0, 3.1 is equivalent to the following usual problem: 0, − ϕ r, u u 0 u0 0, 3.5 where obviously 0 is the unique solution to this problem. Since 0 ∈ Ω, we have proved that 3.4 has no solution u,λ on ∂Ω×0, 1, then we get that, for each λ ∈ 0, 1, Leray-Schauder’s degree dLS I − Ψ·, λ, Ω, 0 is well defined. From the homotopy invariant property of that degree, we have dLS I − Ψu, 1, Ω, 0 dLS I − Ψu, 0, Ω, 0 1. 3.6 This completes the proof. In the following, we will give an application of Theorem 3.1. Denote that σ 4T/4T 1 and Ωε j j u ∈ PC max u u < ε . 0 1≤j≤N 1 3.7 0 Obviously, Ωε is an open subset of PC1 . Assume that H1 fr, u, v δgr, u, v, where δ is a positive parameter, and g is Caratheodory. H2 ki1 |Ai u, v| ≤ σ/2ε, ki1 |Bi u, v| ≤ {min|ε/4T 1|pr /21|Nε|qrpr 1/qr , r∈I 1/2}, for all u, v ∈ RN × RN . Theorem 3.2. If H1 -H2 are satisfied, then problem 1.1–1.4 has at least one solution on Ωε , when positive parameter δ is small enough. Proof. Let one consider the problem u Ψu, λ λ ri <r where Kλ · is defined in 3.3. Ai Kλ u, 3.8 Journal of Inequalities and Applications 9 Obviously, u is a solution of 1.1–1.4 if and only if u is a solution of the abstract equation 3.8 when λ 1. We only need to prove that the conditions of Theorem 3.1 are satisfied. 10 When positive parameter δ is small enough, for any u ∈ Ωε , we can see that the mapping r → δgr, u, u belongs to {u ∈ L1 | uL1 < 1/3}. 20 We shall prove that for each λ ∈ 0, 1 the problem λf r, u, u , ϕ r, u r ∈ 0, T , r / ri , lim ur − lim− ur λAi r → ri r → ri lim ur, lim− u r , r → ri− lim ϕ r, u r − lim− ϕ r, u r λBi r → ri r → ri 3.9 r → ri i 1, . . . , k, lim ur, lim− u r , r → ri− r → ri i 1, . . . , k, u 0 u0 0 has no solution on ∂Ωε . If it is false, then there exists a λ ∈ 0, 1 and u ∈ ∂Ωε is a solution of 3.8. Then there exists an j ∈ {1, . . . , N} such that |uj |0 |uj |0 ε. i Suppose that |uj |0 ≥ σε, then |uj |0 ≤ 1 − σε. For any r ∈ J, since u0 0, according to H2 and 1.2, we have r j j j j u tdt Ai u r u r − u 0 0 0<r <r i T ≤ uj tdt |Ai | 0 ≤ T 0<ri <r 1 − σεdr 0 k 3.10 |Ai | i1 σ 3σ σ ε ε ε. 4 2 4 ≤ It is a contradiction. ii Suppose that |uj |0 < σε, 1−σε < |uj |0 ≤ ε. This implies that |uj r∗ | > 1−σε 1/4T 1ε for some r∗ ∈ J. Denote that |u |pr−1 uj r ϕ r, u r 1/qr . 1 |u |qrpr j 3.11 10 Journal of Inequalities and Applications Since u 0 0, we have r∗ ϕj r∗ , u r∗ λ 0 j f j r, u, u dr λ Bi . 3.12 0<ri <r∗ According to H1 -H2 , when positive parameter δ is small enough, we have r∗ |ε/4T 1|pr∗ j j j ≤ λ λ ≤ , u B r r, u, u r dr f ϕ ∗ ∗ 1/qr∗ 0<r <r i 0 i ∗ 1 |Nε|qr∗ pr∗ ≤δ T 0 βNε rdr k i1 |ε/4T 1|pr∗ |Bi | < 1/qr∗ . qr∗ pr∗ 1 |Nε| 3.13 It is a contradiction. Summarizing this argument, for each λ ∈ 0, 1, the problem 3.8 with 1.4 has no solution on ∂Ωε . Since 0 ∈ Ωε and 0 is the unique solution of u Ψu, 0, then the Leray-Schauder’s degree 0. dLS I − Ψ·, 0, Ω, 0 1 / 3.14 This completes the proof. In the following, we will discuss the existence of nonnegative solutions of 1.1–1.4. For any x {x1 , . . . , xN } ∈ RN , the notation x ≥ 0 x > 0 means that xl ≥ 0 xl > 0 for every l ∈ {1, . . . , N}. Assume the following H3 fr, u, v δgr, u, v, where δ is a positive parameter, and gr, u, v τr |u|q1 r−1 u μr|v|q2 r−1 v γr, 3.15 where q1 , q2 ∈ CJ, R, 0 ≤ q1 r, and 0 ≤ q2 r, for all r ∈ J. j j H4 Ai x, yyj ≥ 0, and Bi x, yyj ≥ 0, for all x, y ∈ RN , i 1, . . . , k, j 1, . . . , N. H5 μ, τ ∈ CJ, R . H6 γ γ 1 , . . . , γ N ∈ CJ, RN satisfies γ0 > 0, t 0 γsds ≥ 0, for all t ∈ 0, T . Theorem 3.3. If H2 –H6 are satisfied, then the problem 1.1–1.4 has a nonnegative solution, when positive parameter δ is small enough. Journal of Inequalities and Applications 11 Proof. From Theorem 3.2, we can get the existence of solutions of 1.1–1.4. If u is a solution of 1.1–1.4, according to 1.4 and H6 , then we have f 0, u0, u 0 δγ0 > 0. 3.16 Obviously ϕ r, u r r q r−1 δ τr |u|q1 r−1 u μru 2 u γr ds, ∀r ∈ 0, r1 . 3.17 0 When r → 0 , we have q r−1 τr |u|q1 r−1 u μru 2 u γr > 0, 3.18 then we can see that there exists a ξ ∈ 0, r1 such that ϕr, u r > 0 when r ∈ 0, ξ. Thus u r > 0 for any r ∈ 0, ξ. Thus ur is increasing in 0, ξ, that is, uη2 ≥ uη1 for any η1 , η2 ∈ 0, ξ with η1 < η2 . Since u0 0, it is easy to see that ur > 0 for any r ∈ 0, ξ. From 3.17 and H5 , we can easily see that ur > 0, u r > 0, lim ur > 0, r → r1− ∀r ∈ 0, r1 , lim u r > 0. 3.19 r → r1− From H4 , we can see that lim ur > 0, r → r1 lim u r > 0. r → r1 3.20 Similarly, we can see that ur > 0, u r > 0, ∀r ∈ r1 , r2 . 3.21 ∀r ∈ J . 3.22 Repeating the step, we can see that ur > 0, u r > 0, Hence u is nonnegative. This completes the proof. Acknowledgments This paper is partly supported by the National Science Foundation of China 10701066, 10926075 and 10971087, China Postdoctoral Science Foundation funded project 20090460969, the Natural Science Foundation of Henan Education Committee 2008-75565, and the Natural Science Foundation of Jiangsu Education Committee 08KJD110007. 12 Journal of Inequalities and Applications References 1 J. Jiao, L. Chen, and L. Li, “Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 458–463, 2008. 2 L. Liu, L. Hu, and Y. Wu, “Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3774–3789, 2008. 3 X. Liu and D. Guo, “Periodic boundary value problems for a class of second-order impulsive integrodifferential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 284–302, 1997. 4 J. Shen and W. Wang, “Impulsive boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 4055–4062, 2008. 5 M. Yao, A. Zhao, and J. Yan, “Periodic boundary value problems of second-order impulsive differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 262–273, 2009. 6 A. Cabada and J. Tomeček, “Extremal solutions for nonlinear functional φ-Laplacian impulsive equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 3, pp. 827–841, 2007. 7 Q. Zhang, X. Liu, and Z. Qiu, “Existence of solutions for weighted pr-Laplacian impulsive system periodic-like boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 3596–3611, 2009. 8 Q. Zhang, Z. Qiu, and X. Liu, “Existence of solutions and nonnegative solutions for weighted pr-Laplacian impulsive system multi-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 3814–3825, 2009. 9 R. Filippucci, “Entire radial solutions of elliptic systems and inequalities of the mean curvature type,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 604–620, 2007. 10 A. Greco, “On the existence of large solutions for equations of prescribed mean curvature,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 4, pp. 571–583, 1998. 11 N. M. Ivochkina, “The Dirichlet problem for the curvature equation of order m,” Leningrad Mathematical Journal, vol. 2, pp. 631–654, 1991. 12 K. E. Lancaster and D. Siegel, “Existence and behavior of the radial limits of a bounded capillary surface at a corner,” Pacific Journal of Mathematics, vol. 176, no. 1, pp. 165–194, 1996. 13 H. Pan, “One-dimensional prescribed mean curvature equation with exponential nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 999–1010, 2009. 14 K. Takimoto, “Solution to the boundary blowup problem for k-curvature equation,” Calculus of Variations and Partial Differential Equations, vol. 26, no. 3, pp. 357–377, 2006. 15 N. S. Trudinger, “The Dirichlet problem for the prescribed curvature equations,” Archive for Rational Mechanics and Analysis, vol. 111, no. 2, pp. 153–179, 1990. 16 Q. Zhang, Z. Qiu, and X. Liu, “Existence of solutions for prescribed variable exponent mean curvature system boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2964–2975, 2009. 17 Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006. 18 M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000. 19 V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Mathematics of the USSR. Izvestiya, vol. 29, pp. 33–36, 1987. 20 X.-L. Fan, H.-Q. Wu, and F.-Z. Wang, “Hartman-type results for pt-Laplacian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 2, pp. 585–594, 2003. 21 X.-L. Fan and D. Zhao, “Regularity of minimum points of variational integrals with continuous pxgrowth conditions,” Chinese Annals of Mathematics. Series A, vol. 17, no. 5, pp. 557–564, 1996. 22 X.-L. Fan and D. Zhao, “On the spaces Lpx Ω and W m,px Ω,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001. 23 X.-L. Fan and Q.-H. Zhang, “Existence of solutions for px-Laplacian Dirichlet problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 8, pp. 1843–1852, 2003. 24 X.-L. Fan, Y. Z. Zhao, and Q. H. Zhang, “A strong maximum principle for px-Laplace equations,” Chinese Journal of Contemporary Mathematics, vol. 24, no. 4, pp. 495–500, 2003. Journal of Inequalities and Applications 13 25 X.-L. Fan, Q. H. Zhang, and D. Zhao, “Eigenvalues of px-Laplacian Dirichlet problem,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 306–317, 2005. 26 A. El Hamidi, “Existence results to elliptic systems with nonstandard growth conditions,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 30–42, 2004. 27 O. Kováčik and J. Rákosnı́k, “On spaces Lpx Ω and W k,px Ω,” Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–618, 1991. 28 P. Marcellini, “Regularity and existence of solutions of elliptic equations with p, q-growth conditions,” Journal of Differential Equations, vol. 90, no. 1, pp. 1–30, 1991. 29 M. Mihăilescu and V. Rădulescu, “Continuous spectrum for a class of nonhomogeneous differential operators,” Manuscripta Mathematica, vol. 125, no. 2, pp. 157–167, 2008. 30 J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983. 31 S. G. Samko, “Density C0∞ RN in the generalized Sobolev spaces W m,px RN ,” Doklady Akademii Nauk. Rossiı̆skaya Akademiya Nauk, vol. 369, no. 4, pp. 451–454, 1999. 32 Q. H. Zhang, “A strong maximum principle for differential equations with nonstandard px-growth conditions,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 24–32, 2005. 33 Q. H. Zhang, “Existence of solutions for px-Laplacian equations with singular coefficients in RN ,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 38–50, 2008. 34 Q. Zhang, X. Liu, and Z. Qiu, “The method of subsuper solutions for weighted pr-Laplacian equation boundary value problems,” Journal of Inequalities and Applications, vol. 2008, Article ID 621621, 18 pages, 2008. 35 Q. Zhang, Z. Qiu, and X. Liu, “Existence of solutions for a class of weighted pt-Laplacian system multipoint boundary value problems,” Journal of Inequalities and Applications, vol. 2008, Article ID 791762, 18 pages, 2008. 36 R. Manásevich and J. Mawhin, “Periodic solutions for nonlinear systems with p-Laplacian-like operators,” Journal of Differential Equations, vol. 145, no. 2, pp. 367–393, 1998.